Bed sheets are commonly used to cover a mattress to provide a bed upon which one can sleep. It is desirable to keep a bed sheet secured on a mattress when in use for a variety of reasons. For example, it is highly desirable to keep the bed sheet on a mattress for safety as applied to crib sheets and the like to prevent the sheet from being removed from the mattress and the infant becoming entangled in the sheet, as is discussed in the present inventor's previous patent, U.S. Pat. No. 6,804,844. There is a continued desire for further improvements as it relates to mattress sheets and different securement mechanisms for different applications, which is the subject of the present invention.